Paradox

paradox,

statement that appears self-contradictory but actually has a basis in truth, e.g., Oscar Wilde's "Ignorance is like a delicate fruit; touch it and the bloom is gone." Many New Critics maintained that paradox is not just a rhetorical or illustrative device but a basic aspect of all poetic language.

Paradox

an unexpected statement (expression, proposition), unusual (if only in form) and differing sharply from the generally accepted, traditional opinion on a particular question. In this sense, the epithet “paradoxical”—that is, “not standard,” “diverging from tradition”—is the opposite of “orthodox,” used as a synonym for “approved,” “generally accepted,” “following prevailing tradition literally.” Every paradox appears to be a negation of a view that seems to be “indisputably correct.” (The truth of this impression is irrelevant.) The very term “paradox” originated in ancient philosophy to describe a new, unusual, or original opinion. Since it is much simpler to perceive the originality of a statement than to ascertain its truth or falsity, Parádoxical statements are often accepted as evidence of the independence and originality of the opinions expressed in them, especially if they are expressed effectively, clearly, and aphoristi-cally.

In some cases, of course, the reputation for originality may be totally deserved. For example, the following philosophical and ethical generalizations are paradoxical: “I disapprove of what you say, but I will defend to the death your right to say it” (Voltaire); “People are cruel, but man is good” (R. Tagore). Regardless of the profundity or truth of a statement, particularly an oral one, its paradoxical character will attract attention. Thus, in addition to a general logical consistency of exposition and a colorful style, the essential attributes of oratory include unexpectedness of conclusions, their failure to follow from the “natural” course of ideas.

However, the opposite effect is often observed: behavior or a statement that contradicts common sense, if only superficially, is characterized as paradoxical, as evidence (in a sense) of the “inconsistency” of a corresponding behavior or statement. In the “paradox of acting,” which was first noted by Diderot, an actor may evoke in the spectators a total illusion of the feelings depicted, even though he himself experiences none of them. The reverse of this paradox has been used by O. Wilde: one of his heroines cannot play the role of Juliet precisely because she herself has fallen in love.

Both tendencies in the treatment of paradoxes are manifested in the effect of the witty and unexpected conclusions of anecdotes. From a more general standpoint, they may underlie the comic as an aesthetic category. In T. Jefferson’s statement, “Victory and defeat are each of the same price,” the paradox consists only in directing attention to something that often escapes notice. If Jefferson’s statement is taken seriously by modern readers, then many statements sound like outright parodies (Shaw’s “Do not do unto others as you would they should do unto you. Their tastes may not be the same”; and Wilde’s “I never put off till tomorrow what I can possibly do the day after”). Paradoxes lie at the root of many proverbs (“The quieter you go, the further you will get”), and they are the foundation for a number of literary genres. For example, “The Magnate,” a well-known fable by I. A. Krylov, is constructed on a paradox: the foolish ruler enters paradise by laziness and idleness. Paradoxes are widely used as an artistic device in “nonsense verses” for children (L. Carroll, A. A. Milne, E. Lear, K. I. Chukovskii).

In logic. Although the scientific interpretation of the term “paradox” grew out of the everyday one, it does not completely coincide with it. Insofar as truth is naturally regarded as the “norm” in science, all deviations from the truth—that is, falsehoods and contradictions—are naturally regarded as paradoxes. Therefore, in logic the term “paradox” is considered a synonym for the terms “antinomy” and “contradiction” and is applied to any reasoning that demonstrates both the truth of a statement and the truth of its negation. In this case, only formally correct conclusions (those corresponding to accepted logical norms) are under consideration, and not reasoning marked by fallacies, either deliberate (sophisms) or accidental (paralogisms). For the various meanings and refinements of the concept of “proof,” there are different meanings (levels) of the concept of “paradox.” At the same time, analysis of any argument that is or claims to be a proof shows that it is based on certain assumptions, either hidden or explicit, either specific to it or characteristic of a theory as a whole. (The assumptions underlying a theory are usually called axioms or postulates.) Thus, the presence of a paradox implies some incompatibility among the assumptions or, in the case of a theory constructed according to the axiomatic method, some contradiction in the system of axioms. However, even if the removal of an assumption leads to the elimination of a specific paradox, it does not in general guarantee the elimination of all paradoxes. On the other hand, the careless rejection of too many assumptions, or of those that are too strong, may result in a substantially weaker theory.

Success in meeting the conditions of consistency and completeness presupposes the painstaking discovery of all the hidden assumptions in the theory under consideration, and subsequently, the explicit listing and formulation of them. At one time, it was thought that these tasks could be accomplished by the axiomatic method, which found its most complete expression in D. Hilbert’s proposal for basing mathematics on logic (meta-mathematics). The first task was considered to be the elimination of the paradoxes discovered at the turn of the century in the theory of sets, which is the foundation for almost all of mathematics. It was thought that this problem could be solved by creating and then proving the consistency of systems of axiomatic set theory suitable for a sufficiently complete construction of mathematical theories. For example, one of the most famous paradoxes in the theory of sets, Russell’s paradox, involves the set R of all sets that are not members of themselves. Such an R is a member of itself if and only if it is not a member of itself. Therefore, the assumption that R is a member of itself leads to the negation of that assumption, from which it follows that R is not a member of itself. This is so even according to the rules of intuitionist logic—that is, without applying the law of the excluded middle. However, if R is not a member of itself, then (based on the initial statement) it is a member of itself. Thus, two mutually contradictory statements have been proved; herein lies the paradox.

The system of axiomatic set theory developed by E. Zermelo and the Zermelo-Fraenkel axiomatic set theory simply dismiss the question of whether R is a member of itself, because the axioms in these systems do not permit consideration of such an R, having denied its existence. With systems devised by J. von Neumann, P. Bernays, and K. Gödel, it is possible to consider such an R. However, as an aggregate of sets, it is declared, by means of restrictive axioms, to be not a set but a class—that is, it is declared in advance that R cannot be a member of anything, including itself. Thus, Russell’s question is again eliminated. Finally, the theory of types, which was developed by A. N. Whitehead (Great Britain) and Russell, has given rise to a number of modified versions, such as the systems proposed by W. V. O. Quine (USA). These modified systems allow for the consideration of any sets described by meaningful linguistic expressions, as well as for any questions relative to them. However, expressions such as “the set of all sets not members of themselves” are declared meaningless because they violate certain conventions of a linguistic (syntactic) character. In a similar manner, the above-mentioned theories eliminate other famous paradoxes in the theory of sets, including G. Cantor’s paradox, in which the cardinal number of the set of all subsets of the “set of all sets” inevitably turns out to be greater than that of the set of all sets.

However, none of the systems of axiomatic set theory can completely solve the problem of eliminating paradoxes, because Hilbert’s program for the foundation of mathematics has turned out to be unattainable. According to Gödel’s theorem (1931), even if they are not contradictory, sufficiently rich axiomatic theories (including formal arithmetic of the natural numbers, and, to a greater degree, axiomatic set theory) cannot be proved consistent using only those methods acceptable under the traditional proof theory developed by Hilbert. In classical mathematics and logic, this restriction was overcome by using stronger methods of mathematical reasoning—methods that were, in a certain sense, constructive, but not “finite” according to Hilbert. Applying these methods, proofs of the consistency of formal arithmetic were obtained (the Soviet mathematician P. S. Novi-kov and the German mathematicians G. Gentzen, W. Acker-mann, and K. Schütte, for example).

The intuitionist and constructivist schools consider it totally unnecessary to deal with the problem of paradoxes. The “effective” methods used by them to construct mathematical theories lead essentially to completely new scientific systems, which begin by banishing the “metaphysical” methods of reasoning and concept formation responsible for the appearance of paradoxes in classical theories. Finally, in the ultraintuitionist program for the foundation of mathematics, the problem of paradoxes is solved by a thorough revision of the very concept of mathematical proof, in order to make it possible to prove the consistency of certain systems of axiomatic set theory (in ultraintuitionist terms, “the unattainability of contradiction”).

The paradoxes under discussion are often called logical Parádoxes, since they may be reformulated in purely logical terms. For example, Russell’s paradox can be restated as follows. Attributes that do not apply to themselves (for example, “blue,” or “foolish”) are impredicative; those that apply to themselves are predicative (for example, “abstract”). The attribute “impredicative” is impredicative if and only if it is predicative.

Some logicians, including the Soviet scholar D. A. Bochvar, consider as “logic proper” (“pure logic”) only a narrow version of the predicate calculus (with equality, perhaps)—a version that is free of paradoxes. From Bochvar’s point of view, paradoxes can still arise in the theory of sets, which includes a broader version of the predicate calculus. This is so because of the unrestricted application of the principle of abstraction, which allows consideration of a set of objects defined by arbitrary properties. The elimination of paradoxes is achieved by means of many-valued logic. In addition to “true” and “false,” paradoxical statements such as Russell’s are assigned a third truth value: “meaningless.”

Another important class of paradoxes (the semantic Parádoxes) also arises in the consideration of certain concepts in the theory of sets and higher-order logic and is associated with the concepts of “denotation,” “nomination,” “truth,” “falsity,” and so forth. Among the semantic paradoxes is the Richard-Berry paradox, one formulation of which involves the phrase “the least natural number not nameable in fewer than 22 syllables.” At least according to the ordinary idea of “definability,” this statement defines a certain natural number in 21 syllables. The oldest known paradox is the paradox of the Liar, or the “lying Cretan,” which is associated with the statement “all Cretans are liars” (attributed to the Cretan philosopher Epimenides), or, on an even simpler level, with the statement “I am lying.” Another semantic paradox is Grelling’s. Adjectives are termed autologi-cal if they apply to themselves (for example, “English” or “polysyllabic”) and heterological if they do not (for example, “Russian,” “monosyllabic,” “yellow,” and “cold”). Consequently, the adjective “heterological” is heterological if and only if it is autological.

Because semantic paradoxes are formulated more in linguistic terms than in logical and mathematical ones, their resolution has not been considered crucial to the foundations of logic and mathematics. Nonetheless, the semantic paradoxes are closely linked with the logical paradoxes: the latter are concerned with concepts, and the former with their names (for example, Russell’s paradox and Grelling’s paradox).

The term “paradox” is also used in logic and mathematics in a broader, almost colloquial sense to refer not to a genuine contradiction but only to a lack of correspondence between formal implications and their intuitive prototypes. For example, the paradoxes of material implication (“anything follows from a falsehood” and “the truth follows from any statement”), which can be proved by the classical logic of propositions, reveal a lack of agreement between everyday and formal logical interpretations of implication. The relative character of the concepts of enumerability and nonenumerability is shown by Skolem’s paradox in axiomatic set theory, according to which the concept of a nonenumerable set may be expressed by means of an enumerable model. Similar paradoxes are encountered in ethics and in modal logic (the lack of correspondence between the modalities “possible” and “necessary” and their formal axiomatic descriptions).

The distinction drawn earlier in this article between paradoxes (conclusions based on formally “correct” reasoning) and sophisms (based on deliberately fallacious reasoning) is, to a significant degree, conventional. Many arguments traditionally classified as sophisms and “pseudoparadoxes” turn out to be very important in new logical and methodological trends. For example, the ancient paradox of the Heap (one seed is not a heap; the addition of one seed does not produce a heap; but a million seeds constitute a heap), which has also been formulated as the paradox of the Bald Man, had until recently been “resolved” by a simple reference to the insufficient definition of the concept of “heap.” However, among the important original ideas of the ultraintuitionist approach are the conscious rejection of this type of direct “resolution” and the clarification of the possibility for the precise use of concepts such as “more.” The concepts of “antimony” and “aporia” are closely related to the concept of “paradox.”

In science. Paradoxes—conclusions that are drawn from apparently correct (in any case, generally accepted) first principles but that are found to contradict experiment (and, perhaps, intuition and common sense)—are encountered not only in the purely deductive sciences but also, for example, in physics. The theories of relativity and quantum mechanics abound in “paradoxical” conclusions that contradict centuries of scientific tradition. The analysis of paradoxes (for example, in physics and cosmogony, the photometric and gravitational paradoxes, also known as the cosmological paradoxes) has played an important role in the sciences, just as it has in logic and in mathematics. In a broader sense, the same may be said of any refinements of scientific theories arising from contradictions between new results and principles previously thought to be firmly established. Such refinements are an integral part of the general development of science.

Paradox

A relational database management (DBMS) and application development system for Windows from Corel. It includes the PAL programming language for writing complex business applications. When Paradox was originally released under DOS, it was noted for its visual query by example method, which made asking questions much easier than comparable products of the time. Originally developed by Ansa Software, it was later acquired by Borland and then Corel. The product was named Corel Paradox for a while, then offered as part of Corel's WordPerfect Office suites.

An Early Paradox Query

In the mid-1980s, Paradox was the first DBMS on a PC that made linking tables easier. The ability to associate relationships by typing sample words was a breakthrough for that time. The Customer No. and Part No. fields are linked by pressing a function key and typing in the common words "ABC" and "XYZ. Any words suffice as long as they are the same.

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